Primitive of x squared over x squared minus a squared/Logarithm Form

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Theorem

$\ds \int \frac {x^2 \rd x} {x^2 - a^2} = x + \frac a 2 \map \ln {\frac {x - a} {x + a} } + C$

for $x^2 > a^2$.


Proof

Let:

\(\ds \int \frac {x^2 \rd x} {x^2 - a^2}\) \(=\) \(\ds \int \frac {x^2 - a^2 + a^2} {x^2 - a^2} \rd x\)
\(\ds \) \(=\) \(\ds \int \frac {x^2 - a^2} {x^2 - a^2} \rd x + \int \frac {a^2} {x^2 - a^2} \rd x\)
\(\ds \) \(=\) \(\ds \int \d x + a^2 \int \frac {\d x} {x^2 - a^2}\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds x + a^2 \int \frac {\d x} {x^2 - a^2} + C\) Primitive of Constant
\(\ds \) \(=\) \(\ds x + a^2 \paren {\frac 1 {2 a} \map \ln {\frac {x - a} {x + a} } } + C\) Primitive of $\dfrac 1 {x^2 - a^2}$: Logarithm Form
\(\ds \) \(=\) \(\ds x + \frac a 2 \map \ln {\frac {x - a} {x + a} } + C\) simplifying

$\blacksquare$


Sources

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